23 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. |
23 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. |
24 |
24 |
25 We of course define $\bc_0(X) = \lf(X)$. |
25 We of course define $\bc_0(X) = \lf(X)$. |
26 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. |
26 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. |
27 We'll omit this sort of detail in the rest of this section.) |
27 We'll omit this sort of detail in the rest of this section.) |
28 In other words, $\bc_0(X)$ is just the vector space of all (linearized) fields on $X$. |
28 In other words, $\bc_0(X)$ is just the vector space of fields on $X$. |
29 |
29 |
30 $\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$. |
30 We want the vector space $\bc_1(X)$ to capture `the space of all local relations that can be imposed on $\bc_0(X)$'. |
31 Less roughly (but still not the official definition), $\bc_1(X)$ is finite linear |
31 Thus we say a $1$-blob diagram consists of |
32 combinations of 1-blob diagrams, where a 1-blob diagram consists of |
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33 \begin{itemize} |
32 \begin{itemize} |
34 \item An embedded closed ball (``blob") $B \sub X$. |
33 \item An embedded closed ball (``blob") $B \sub X$. |
35 \item A field $r \in \cC(X \setmin B; c)$ |
34 \item A boundary condition $c \in \cC(\bdy B) = \cC(\bd(X \setmin B))$. |
36 (for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). |
35 \item A field $r \in \cC(X \setmin B; c)$. |
37 \item A local relation field $u \in U(B; c)$ |
36 \item A local relation field $u \in U(B; c)$. |
38 (same $c$ as previous bullet). |
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39 \end{itemize} |
37 \end{itemize} |
40 (See Figure \ref{blob1diagram}.) |
38 (See Figure \ref{blob1diagram}.) |
41 \begin{figure}[t]\begin{equation*} |
39 \begin{figure}[t]\begin{equation*} |
42 \mathfig{.6}{definition/single-blob} |
40 \mathfig{.6}{definition/single-blob} |
43 \end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure} |
41 \end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure} |
44 In order to get the linear structure correct, we (officially) define |
42 In order to get the linear structure correct, the actual definition is |
45 \[ |
43 \[ |
46 \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . |
44 \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . |
47 \] |
45 \] |
48 The first direct sum is indexed by all blobs $B\subset X$, and the second |
46 The first direct sum is indexed by all blobs $B\subset X$, and the second |
49 by all boundary conditions $c \in \cC(\bd B)$. |
47 by all boundary conditions $c \in \cC(\bd B)$. |
59 (but keeping the blob label $u$). |
57 (but keeping the blob label $u$). |
60 |
58 |
61 Note that the skein space $A(X)$ |
59 Note that the skein space $A(X)$ |
62 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. This is Property \ref{property:skein-modules}, and also used in the second half of Property \ref{property:contractibility}. |
60 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. This is Property \ref{property:skein-modules}, and also used in the second half of Property \ref{property:contractibility}. |
63 |
61 |
64 $\bc_2(X)$ is, roughly, the space of all relations (redundancies, syzygies) among the |
62 Next, we want the vector space $\bc_2(X)$ to capture `the space of all relations (redundancies, syzygies) among the |
65 local relations encoded in $\bc_1(X)$. |
63 local relations encoded in $\bc_1(X)$'. |
66 More specifically, $\bc_2(X)$ is the space of all finite linear combinations of |
64 More specifically, a $2$-blob diagram, comes in one of two types, disjoint and nested. |
67 2-blob diagrams, of which there are two types, disjoint and nested. |
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68 |
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69 A disjoint 2-blob diagram consists of |
65 A disjoint 2-blob diagram consists of |
70 \begin{itemize} |
66 \begin{itemize} |
71 \item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors. |
67 \item A pair of closed balls (blobs) $B_1, B_2 \sub X$ with disjoint interiors. |
72 \item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$ |
68 \item A field $r \in \cC(X \setmin (B_1 \cup B_2); c_1, c_2)$ |
73 (where $c_i \in \cC(\bd B_i)$). |
69 (where $c_i \in \cC(\bd B_i)$). |
74 \item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. \nn{We're inconsistent with the indexes -- are they 0,1 or 1,2? I'd prefer 1,2.} |
70 \item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. |
75 \end{itemize} |
71 \end{itemize} |
76 (See Figure \ref{blob2ddiagram}.) |
72 (See Figure \ref{blob2ddiagram}.) |
77 \begin{figure}[t]\begin{equation*} |
73 \begin{figure}[t]\begin{equation*} |
78 \mathfig{.6}{definition/disjoint-blobs} |
74 \mathfig{.6}{definition/disjoint-blobs} |
79 \end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure} |
75 \end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure} |
80 We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$; |
76 We also identify $(B_1, B_2, u_1, u_2, r)$ with $-(B_2, B_1, u_2, u_1, r)$; |
81 reversing the order of the blobs changes the sign. |
77 reversing the order of the blobs changes the sign. |
82 Define $\bd(B_0, B_1, u_0, u_1, r) = |
78 Define $\bd(B_1, B_2, u_1, u_2, r) = |
83 (B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$. |
79 (B_2, u_2, u_1\bullet r) - (B_1, u_1, u_2\bullet r) \in \bc_1(X)$. |
84 In other words, the boundary of a disjoint 2-blob diagram |
80 In other words, the boundary of a disjoint 2-blob diagram |
85 is the sum (with alternating signs) |
81 is the sum (with alternating signs) |
86 of the two ways of erasing one of the blobs. |
82 of the two ways of erasing one of the blobs. |
87 It's easy to check that $\bd^2 = 0$. |
83 It's easy to check that $\bd^2 = 0$. |
88 |
84 |
89 A nested 2-blob diagram consists of |
85 A nested 2-blob diagram consists of |
90 \begin{itemize} |
86 \begin{itemize} |
91 \item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. |
87 \item A pair of nested balls (blobs) $B_1 \sub B_2 \sub X$. |
92 \item A field $r \in \cC(X \setmin B_0; c_0)$ |
88 \item A field $r' \in \cC(B_2 \setminus B_1; c_1, c_2)$ (for some $c_1 \in \cC(\bdy B_1)$ and $c_2 \in \cC(\bdy B_2)$). |
93 (for some $c_0 \in \cC(\bd B_0)$), which is splittable along $\bd B_1$. |
89 \item A field $r \in \cC(X \setminus B_2; c_2)$. |
94 \item A local relation field $u_0 \in U(B_0; c_0)$. |
90 \item A local relation field $u \in U(B_1; c_1)$. |
95 \end{itemize} |
91 \end{itemize} |
96 (See Figure \ref{blob2ndiagram}.) |
92 (See Figure \ref{blob2ndiagram}.) |
97 \begin{figure}[t]\begin{equation*} |
93 \begin{figure}[t]\begin{equation*} |
98 \mathfig{.6}{definition/nested-blobs} |
94 \mathfig{.6}{definition/nested-blobs} |
99 \end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure} |
95 \end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure} |
100 Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
96 Define $\bd(B_1, B_2, u, r', r) = (B_2, u\bullet r', r) - (B_1, u, r' \bullet r)$. |
101 (for some $c_1 \in \cC(B_1)$) and |
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102 $r' \in \cC(X \setmin B_1; c_1)$. |
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103 Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$. |
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104 Note that the requirement that |
97 Note that the requirement that |
105 local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$. |
98 local relations are an ideal with respect to gluing guarantees that $u\bullet r' \in U(B_2)$. |
106 As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
99 As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
107 sum of the two ways of erasing one of the blobs. |
100 sum of the two ways of erasing one of the blobs. |
108 If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$. |
101 When we erase the inner blob, the outer blob inherits the label $u\bullet r'$. |
109 It is again easy to check that $\bd^2 = 0$. |
102 It is again easy to check that $\bd^2 = 0$. |
110 |
103 |
111 As with the 1-blob diagrams, in order to get the linear structure correct it is better to define |
104 As with the $1$-blob diagrams, in order to get the linear structure correct the actual definition is |
112 (officially) |
|
113 \begin{eqnarray*} |
105 \begin{eqnarray*} |
114 \bc_2(X) & \deq & |
106 \bc_2(X) & \deq & |
115 \left( |
107 \left( |
116 \bigoplus_{B_0, B_1 \text{disjoint}} \bigoplus_{c_0, c_1} |
108 \bigoplus_{B_1, B_2 \text{disjoint}} \bigoplus_{c_1, c_2} |
117 U(B_0; c_0) \otimes U(B_1; c_1) \otimes \lf(X\setmin (B_0\cup B_1); c_0, c_1) |
109 U(B_1; c_1) \otimes U(B_2; c_2) \otimes \lf(X\setmin (B_1\cup B_2); c_1, c_2) |
118 \right) \\ |
110 \right) \\ |
119 && \bigoplus \left( |
111 && \bigoplus \left( |
120 \bigoplus_{B_0 \subset B_1} \bigoplus_{c_0} |
112 \bigoplus_{B_1 \subset B_2} \bigoplus_{c_1, c_2} |
121 U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0) |
113 U(B_1; c_1) \otimes \lf(B_2 \setmin B_1; c_1) \tensor \cC(X \setminus B_2; c_2) |
122 \right) . |
114 \right) . |
123 \end{eqnarray*} |
115 \end{eqnarray*} |
124 The final $\lf(X\setmin B_0; c_0)$ above really means fields splittable along $\bd B_1$, |
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125 but we didn't feel like introducing a notation for that. |
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126 For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign |
116 For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign |
127 (rather than a new, linearly independent 2-blob diagram). |
117 (rather than a new, linearly independent 2-blob diagram). \nn{Hmm, I think we should be doing this for nested blobs too -- we shouldn't force the linear indexing of the blobs to have anything to do with the partial ordering by inclusion -- this is what happens below} |
128 |
118 |
129 Now for the general case. |
119 Now for the general case. |
130 A $k$-blob diagram consists of |
120 A $k$-blob diagram consists of |
131 \begin{itemize} |
121 \begin{itemize} |
132 \item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$. |
122 \item A collection of blobs $B_i \sub X$, $i = 1, \ldots, k$. |
133 For each $i$ and $j$, we require that either $B_i$ and $B_j$ have disjoint interiors or |
123 For each $i$ and $j$, we require that either $B_i$ and $B_j$ have disjoint interiors or |
134 $B_i \sub B_j$ or $B_j \sub B_i$. |
124 $B_i \sub B_j$ or $B_j \sub B_i$. |
135 (The case $B_i = B_j$ is allowed. |
125 (The case $B_i = B_j$ is allowed. |
136 If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
126 If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
137 If a blob has no other blobs strictly contained in it, we call it a twig blob. |
127 If a blob has no other blobs strictly contained in it, we call it a twig blob. |